**3. Results**

Any object moving through a fluid will experience an aerodynamic drag that is produced by both pressure and shear forces acting on its surface.

$$D\_t = D\_p + D\_{f'} \tag{15}$$

where *Dt* is the total drag, *Dp* is the pressure drag, and *Df* is the friction drag. Pressure drag is strongly dependent on the shape or form of the object; friction drag is a function of the wall shear stress, which is affected by surface roughness and the Reynolds number [27]. The drag coefficient is defined by the following equations:

$$\mathcal{C}\_{D} = \frac{D\_{t}}{\frac{1}{2}\rho v\_{P}^{2}A} \quad \text{for calculating total drag coefficient} \tag{16}$$

$$\mathbb{C}\_{D\_p} = \frac{D\_p}{\frac{1}{2}\rho v\_p^2 A} \quad \text{for calculating pressure drag coefficient} \tag{17}$$

$$\mathcal{C}\_{D\_f} = \frac{D\_f}{\frac{1}{2}\rho v\_P^2 A} \quad \text{for calculating friction drag coefficient} \tag{18}$$

where *A* is the frontal area, which is the cross-sectional area of the pod calculated by π*d*<sup>2</sup>*po<sup>d</sup>*/4. Thus, in this study, the reference area in Equations (16)–(18) is approximately 7.065 m2. ρ is the reference density of air in the tube pressure and temperature of 300 K. *vP* is the pod speed.

This study mainly focuses on the influence of various parameters on the aerodynamic drag. The amplitude and speed of pressure waves are also presented. The main variables considered here include BR, operating pod speed, internal tube pressure, and pod length. Owing to a fixed cross-sectional area of the pod, the variation in BR is obtained by changing the tube diameter to 6 m for BR = 0.25 (5 m for BR = 0.36). The pod speed was varied from 100 to 350 m/s in intervals of 50 m/s, in other words, 100, 150, 200, 250, 300, 350 m/s. For investigating the flows near the critical Mach number, two more pod speeds were chosen, namely, 225 and 275 m/s. With changing tube pressure, the pod speed of 300 m/s was selected to evaluate four different tube pressures, namely, 101.325, 500, 750, and 1013.25 Pa under a BR of 0.36 and a pod length of 43 m. The Hyperloop Alpha documents used a pod length of 43 m to carry 28 passengers per trip. To investigate the effect of pod length on drag coefficient, the case of *Lre f* = 43 m and two reference cases *Lre f* /2 and 2*Lre f* were selected.

### *3.1. E*ff*ect of Blockage Ratio and Pod Speed*

As indicated in Figure 6, the changes of friction and pressure drag coefficient vary by two values of BR from *vP* of 100 to 350 m/s. The variation of friction and pressure drag is given in insets. Besides, Figures A2 and A3 (Appendix B) also show the variation of friction and pressure drag coefficients on *Re* and Mach number. In Figure 6, at the same *vP*, the friction drag coefficient *CDf* and the pressure drag coefficient *CDp* increase as the BR increases. When the BR is higher, the area where the flow passes through decreases. Thus, the flow becomes harder to bypass, increasing the friction drag generated along the pod surface and resulting in *CDf* increases as BR increases (Figure 6a). It should be noted that from the *vP* of 200 m/s, the effect of BR on *CDf*reduces due to severe choking.

(**b**) Pressure drag coefficient

**Figure 6.** Variations of (**a**) friction and (**b**) pressure drag coefficients with pod speed and blockage ratio. Insets illustrate the variation of drag.

The change in pressure drag depends on the pressure difference between the nose and tail of the pod. With increasing BR, the air becomes more compressible, which increases the pressure magnitude along the pod surface as well as the pressure difference between the nose and the tail, resulting in an increase in pressure drag; hence *CDp* increases. With increasing *vP*, pressure and friction drag increase continuously and significantly. Unlike drag, the drag coefficients witness different tendencies. As shown in Figure 6b, at BR = 0.36, *CDp* increases and reaches the maximum at a *vP* of 225 m/s. Subsequently, it exhibits a continuous drop. By contrast, *CDf* presents an incessant decrease with the increment of *vP*. The results show tendencies similar with those in previous literature [5,11,23,28]. Note that, at lower values of BR, the maximum value of *CDp* changes. In this study, at BR = 0.25, *CDp* reaches a maximum at a *vP* of 250 m/s, which will be compared later with BR = 0.36.

Generally, aerodynamic drag is composed of pressure and friction drags. Unlike in the open air, the Hyperloop pod moves in a tunnel, experiencing more aerodynamic drag owing to the increase in pressure generated by its interaction with the tunnel walls [29]. At higher speed, there will be higher pressure. When the pod passes through the tube at a high speed, a high-pressure region is formed in the front of the pod nose. Meanwhile, the pod tail experiences an increase in the velocity of flow and reduces the pressure behind the pod. This phenomenon is similar to the behavior of flow through a convergent–divergent nozzle and results in a greater pressure difference between the nose and the

tail of the pod; this leads to an increase in the pressure drag. Besides, the low pressure in the tube reduces the friction drag in the Hyperloop system. Hence, in the pod–tube system, pressure drag is more dominant than friction drag.

To evaluate the portions of pressure and friction drag in the total drag, the ratios of the two drag components to the total drag are shown in Figure 7. At BR = 0.36 and *vP* = 100 m/s, pressure drag dominates the total drag by 70%; the remaining percentage belongs to friction drag. However, pressure drag becomes more dominant as *vP* increases owing to the increase of the di fference in pressure at the nose and tail; the friction drag does not change much. This results in the pressure drag becoming more dominant at *vP* of 350 m/s, making up 89.2% of the total drag, whereas friction drag only makes up 10.8%. The portion of *Dp*/*Dt* decreases with reducing BR, whereas the portion of *Df* /*Dt* increases. This is because the e ffect of BR on the pressure drag is much higher than on friction drag. As BR reduces from 0.36 to 0.25, and at *vP* = 100 m/s, pressure drag decreases by more than 50%, whereas the friction decreases by approximately 25%. This gap is greater at *vP* = 350 m/s, where the margins of decrease for pressure drag and friction drag are 22.4% and 6.9%, respectively. Note that the ratios of the pressure and friction drags to the total drag begin to converge from a *vP* of 250 m/s.

**Figure 7.** Ratios of the pressure and friction drags to the total drag. The results are similar to those of Oh et al. (2019) [5], who conducted steady-state simulations.

The pressure contours in Figure 8 show that the pressure in front of the pod increases with increase of BR. Figure 8d,e,j,k reveals shockwaves at the rear end of the pod; these shockwaves are made of normal and oblique shockwaves generated by the interaction and reflection of pressure waves between the tube walls and pod surface. When the local flow velocity exceeds the speed of sound, the interaction and connection of the airflow and pod surface create oblique shockwaves [28]. With increasing *vP*, the shockwaves become more distinct, expand backward, are reflected by the upper and lower walls of the tube [16], and are then weakened downstream owing to the friction of the airflow. This oblique shockwave intensifies the airflow pressure at the rear end of the pod, resulting in an abrupt pressure increase at the tail (Figure 9a). Note that at lower BR, the formation of oblique shockwaves is delayed. The existence of shockwaves a ffects the variation of *CDp* . In this study, shockwaves are noticed before the *vP* where *CDp* is maximized. At BR = 0.25, the shockwave structure can be observed from a *vP* of 225 m/s (Figure 8c), and from 200 m/s at BR = 0.36 (Figure 8h). The previous study conducted by Kim et al. (2011) [12] also indicated that the impact of shockwaves was reduced when BR was reduced. Hence, to limit the e ffect of shockwaves, we considered the influence of BR [8,12].

With increasing BR, the percentage of pressure drag to total drag increases. To explain this phenomenon, Figure 9a illustrates the pressure di fference between the nose and the tail of the pod, which is one of the factors a ffecting pressure drag. As the *vP* increases, the pressure at the nose increases significantly, which is in contrast with the slight decrease in pressure at the tail. This di fference causes

a substantial increase in the pressure difference between the nose and the tail, resulting in a sharp increase in the pressure drag. This pressure difference in turn increases with the increase of *vP* and BR. At the same *vP*, the pressure at the nose with BR = 0.36 is higher than in the case when BR = 0.25 because of the generation of stronger compression waves. Otherwise, the tail pressure experiences a smaller pressure drop when the BR is smaller, because the air in the larger tube expands easily [11,12,30].

Figure 9b illustrates the pressure variation across the pod surface. There is a sudden increase in the pressure at the tail from a *vP* of 250 m/s owing to the formation of strong oblique shockwaves. A noticeable increase in the surface pressure occurs from 200 to 250 m/s, where the choking flow becomes severe. Note that, although an increment in the surface pressure was observed from 100 to 250 m/s, the pressure magnitude at the tail exhibited a considerable drop owing to the stronger flow expansion with increasing *vP*. As oblique shockwaves appear at the tail from a *vP* of 250 m/s, the tail pressure slightly increases.

**Figure 8.** Pressure contours for selected pod speeds with *Ptube* = 101.325 Pa, *Lpod* = 43 m, and (**<sup>a</sup>**–**<sup>e</sup>**) for BR = 0.25, (**f**–**j**) for BR = 0.36. Contour levels are fixed for each pod speed.

(**b**) Pressure distribution on pod surface 

**Figure 9.** (**a**) Pressure difference between the nose and the tail. (**b**) Pressure distribution across the pod surface -BR = 0.36, *Ptube* = 101.325 Pa, *Lpod* = 43 m. Inset shows the magnified view of the tail.
